We have already explained the number of the figures, the character and
number of the premisses, when and how a syllogism is formed; further what
we must look for when a refuting and establishing propositions, and how
we should investigate a given problem in any branch of inquiry, also by
what means we shall obtain principles appropriate to each subject. Since
some syllogisms are universal, others particular, all the universal syllogisms
give more than one result, and of particular syllogisms the affirmative
yield more than one, the negative yield only the stated conclusion. For
all propositions are convertible save only the particular negative: and
the conclusion states one definite thing about another definite thing.
Consequently all syllogisms save the particular negative yield more than
one conclusion, e.g. if A has been proved to to all or to some B, then
B must belong to some A: and if A has been proved to belong to no B, then
B belongs to no A. This is a different conclusion from the former. But
if A does not belong to some B, it is not necessary that B should not belong
to some A: for it may possibly belong to all A.

This then is the reason common to all syllogisms whether universal
or particular. But it is possible to give another reason concerning those
which are universal. For all the things that are subordinate to the middle
term or to the conclusion may be proved by the same syllogism, if the former
are placed in the middle, the latter in the conclusion; e.g. if the conclusion
AB is proved through C, whatever is subordinate to B or C must accept the
predicate A: for if D is included in B as in a whole, and B is included
in A, then D will be included in A. Again if E is included in C as in a
whole, and C is included in A, then E will be included in A. Similarly
if the syllogism is negative. In the second figure it will be possible
to infer only that which is subordinate to the conclusion, e.g. if A belongs
to no B and to all C; we conclude that B belongs to no C. If then D is
subordinate to C, clearly B does not belong to it. But that B does not
belong to what is subordinate to A is not clear by means of the syllogism.
And yet B does not belong to E, if E is subordinate to A. But while it
has been proved through the syllogism that B belongs to no C, it has been
assumed without proof that B does not belong to A, consequently it does
not result through the syllogism that B does not belong to
E.

But in particular syllogisms there will be no necessity of inferring
what is subordinate to the conclusion (for a syllogism does not result
when this premiss is particular), but whatever is subordinate to the middle
term may be inferred, not however through the syllogism, e.g. if A belongs
to all B and B to some C. Nothing can be inferred about that which is subordinate
to C; something can be inferred about that which is subordinate to B, but
not through the preceding syllogism. Similarly in the other figures. That
which is subordinate to the conclusion cannot be proved; the other subordinate
can be proved, only not through the syllogism, just as in the universal
syllogisms what is subordinate to the middle term is proved (as we saw)
from a premiss which is not demonstrated: consequently either a conclusion
is not possible in the case of universal syllogisms or else it is possible
also in the case of particular syllogisms.

Part 2

It is possible for the premisses of the syllogism to be true, or
to be false, or to be the one true, the other false. The conclusion is
either true or false necessarily. From true premisses it is not possible
to draw a false conclusion, but a true conclusion may be drawn from false
premisses, true however only in respect to the fact, not to the reason.
The reason cannot be established from false premisses: why this is so will
be explained in the sequel.

First then that it is not possible to draw a false conclusion from
true premisses, is made clear by this consideration. If it is necessary
that B should be when A is, it is necessary that A should not be when B
is not. If then A is true, B must be true: otherwise it will turn out that
the same thing both is and is not at the same time. But this is impossible.
Let it not, because A is laid down as a single term, be supposed that it
is possible, when a single fact is given, that something should necessarily
result. For that is not possible. For what results necessarily is the conclusion,
and the means by which this comes about are at the least three terms, and
two relations of subject and predicate or premisses. If then it is true
that A belongs to all that to which B belongs, and that B belongs to all
that to which C belongs, it is necessary that A should belong to all that
to which C belongs, and this cannot be false: for then the same thing will
belong and not belong at the same time. So A is posited as one thing, being
two premisses taken together. The same holds good of negative syllogisms:
it is not possible to prove a false conclusion from true
premisses.

But from what is false a true conclusion may be drawn, whether
both the premisses are false or only one, provided that this is not either
of the premisses indifferently, if it is taken as wholly false: but if
the premiss is not taken as wholly false, it does not matter which of the
two is false. (1) Let A belong to the whole of C, but to none of the Bs,
neither let B belong to C. This is possible, e.g. animal belongs to no
stone, nor stone to any man. If then A is taken to belong to all B and
B to all C, A will belong to all C; consequently though both the premisses
are false the conclusion is true: for every man is an animal. Similarly
with the negative. For it is possible that neither A nor B should belong
to any C, although A belongs to all B, e.g. if the same terms are taken
and man is put as middle: for neither animal nor man belongs to any stone,
but animal belongs to every man. Consequently if one term is taken to belong
to none of that to which it does belong, and the other term is taken to
belong to all of that to which it does not belong, though both the premisses
are false the conclusion will be true. (2) A similar proof may be given
if each premiss is partially false.

(3) But if one only of the premisses is false, when the first premiss
is wholly false, e.g. AB, the conclusion will not be true, but if the premiss
BC is wholly false, a true conclusion will be possible. I mean by 'wholly
false' the contrary of the truth, e.g. if what belongs to none is assumed
to belong to all, or if what belongs to all is assumed to belong to none.
Let A belong to no B, and B to all C. If then the premiss BC which I take
is true, and the premiss AB is wholly false, viz. that A belongs to all
B, it is impossible that the conclusion should be true: for A belonged
to none of the Cs, since A belonged to nothing to which B belonged, and
B belonged to all C. Similarly there cannot be a true conclusion if A belongs
to all B, and B to all C, but while the true premiss BC is assumed, the
wholly false premiss AB is also assumed, viz. that A belongs to nothing
to which B belongs: here the conclusion must be false. For A will belong
to all C, since A belongs to everything to which B belongs, and B to all
C. It is clear then that when the first premiss is wholly false, whether
affirmative or negative, and the other premiss is true, the conclusion
cannot be true.

(4) But if the premiss is not wholly false, a true conclusion is
possible. For if A belongs to all C and to some B, and if B belongs to
all C, e.g. animal to every swan and to some white thing, and white to
every swan, then if we take as premisses that A belongs to all B, and B
to all C, A will belong to all C truly: for every swan is an animal. Similarly
if the statement AB is negative. For it is possible that A should belong
to some B and to no C, and that B should belong to all C, e.g. animal to
some white thing, but to no snow, and white to all snow. If then one should
assume that A belongs to no B, and B to all C, then will belong to no
C.

(5) But if the premiss AB, which is assumed, is wholly true, and
the premiss BC is wholly false, a true syllogism will be possible: for
nothing prevents A belonging to all B and to all C, though B belongs to
no C, e.g. these being species of the same genus which are not subordinate
one to the other: for animal belongs both to horse and to man, but horse
to no man. If then it is assumed that A belongs to all B and B to all C,
the conclusion will be true, although the premiss BC is wholly false. Similarly
if the premiss AB is negative. For it is possible that A should belong
neither to any B nor to any C, and that B should not belong to any C, e.g.
a genus to species of another genus: for animal belongs neither to music
nor to the art of healing, nor does music belong to the art of healing.
If then it is assumed that A belongs to no B, and B to all C, the conclusion
will be true.

(6) And if the premiss BC is not wholly false but in part only,
even so the conclusion may be true. For nothing prevents A belonging to
the whole of B and of C, while B belongs to some C, e.g. a genus to its
species and difference: for animal belongs to every man and to every footed
thing, and man to some footed things though not to all. If then it is assumed
that A belongs to all B, and B to all C, A will belong to all C: and this
ex hypothesi is true. Similarly if the premiss AB is negative. For it is
possible that A should neither belong to any B nor to any C, though B belongs
to some C, e.g. a genus to the species of another genus and its difference:
for animal neither belongs to any wisdom nor to any instance of 'speculative',
but wisdom belongs to some instance of 'speculative'. If then it should
be assumed that A belongs to no B, and B to all C, will belong to no C:
and this ex hypothesi is true.

In particular syllogisms it is possible when the first premiss
is wholly false, and the other true, that the conclusion should be true;
also when the first premiss is false in part, and the other true; and when
the first is true, and the particular is false; and when both are false.
(7) For nothing prevents A belonging to no B, but to some C, and B to some
C, e.g. animal belongs to no snow, but to some white thing, and snow to
some white thing. If then snow is taken as middle, and animal as first
term, and it is assumed that A belongs to the whole of B, and B to some
C, then the premiss BC is wholly false, the premiss BC true, and the conclusion
true. Similarly if the premiss AB is negative: for it is possible that
A should belong to the whole of B, but not to some C, although B belongs
to some C, e.g. animal belongs to every man, but does not follow some white,
but man belongs to some white; consequently if man be taken as middle term
and it is assumed that A belongs to no B but B belongs to some C, the conclusion
will be true although the premiss AB is wholly false. (If the premiss AB
is false in part, the conclusion may be true. For nothing prevents A belonging
both to B and to some C, and B belonging to some C, e.g. animal to something
beautiful and to something great, and beautiful belonging to something
great. If then A is assumed to belong to all B, and B to some C, the a
premiss AB will be partially false, the premiss BC will be true, and the
conclusion true. Similarly if the premiss AB is negative. For the same
terms will serve, and in the same positions, to prove the
point.

(9) Again if the premiss AB is true, and the premiss BC is false,
the conclusion may be true. For nothing prevents A belonging to the whole
of B and to some C, while B belongs to no C, e.g. animal to every swan
and to some black things, though swan belongs to no black thing. Consequently
if it should be assumed that A belongs to all B, and B to some C, the conclusion
will be true, although the statement Bc is false. Similarly if the premiss
AB is negative. For it is possible that A should belong to no B, and not
to some C, while B belongs to no C, e.g. a genus to the species of another
genus and to the accident of its own species: for animal belongs to no
number and not to some white things, and number belongs to nothing white.
If then number is taken as middle, and it is assumed that A belongs to
no B, and B to some C, then A will not belong to some C, which ex hypothesi
is true. And the premiss AB is true, the premiss BC
false.

(10) Also if the premiss AB is partially false, and the premiss
BC is false too, the conclusion may be true. For nothing prevents A belonging
to some B and to some C, though B belongs to no C, e.g. if B is the contrary
of C, and both are accidents of the same genus: for animal belongs to some
white things and to some black things, but white belongs to no black thing.
If then it is assumed that A belongs to all B, and B to some C, the conclusion
will be true. Similarly if the premiss AB is negative: for the same terms
arranged in the same way will serve for the proof.

(11) Also though both premisses are false the conclusion may be
true. For it is possible that A may belong to no B and to some C, while
B belongs to no C, e.g. a genus in relation to the species of another genus,
and to the accident of its own species: for animal belongs to no number,
but to some white things, and number to nothing white. If then it is assumed
that A belongs to all B and B to some C, the conclusion will be true, though
both premisses are false. Similarly also if the premiss AB is negative.
For nothing prevents A belonging to the whole of B, and not to some C,
while B belongs to no C, e.g. animal belongs to every swan, and not to
some black things, and swan belongs to nothing black. Consequently if it
is assumed that A belongs to no B, and B to some C, then A does not belong
to some C. The conclusion then is true, but the premisses arc
false.

Part 3

In the middle figure it is possible in every way to reach a true
conclusion through false premisses, whether the syllogisms are universal
or particular, viz. when both premisses are wholly false; when each is
partially false; when one is true, the other wholly false (it does not
matter which of the two premisses is false); if both premisses are partially
false; if one is quite true, the other partially false; if one is wholly
false, the other partially true. For (1) if A belongs to no B and to all
C, e.g. animal to no stone and to every horse, then if the premisses are
stated contrariwise and it is assumed that A belongs to all B and to no
C, though the premisses are wholly false they will yield a true conclusion.
Similarly if A belongs to all B and to no C: for we shall have the same
syllogism.

(2) Again if one premiss is wholly false, the other wholly true:
for nothing prevents A belonging to all B and to all C, though B belongs
to no C, e.g. a genus to its co-ordinate species. For animal belongs to
every horse and man, and no man is a horse. If then it is assumed that
animal belongs to all of the one, and none of the other, the one premiss
will be wholly false, the other wholly true, and the conclusion will be
true whichever term the negative statement concerns.

(3) Also if one premiss is partially false, the other wholly true.
For it is possible that A should belong to some B and to all C, though
B belongs to no C, e.g. animal to some white things and to every raven,
though white belongs to no raven. If then it is assumed that A belongs
to no B, but to the whole of C, the premiss AB is partially false, the
premiss AC wholly true, and the conclusion true. Similarly if the negative
statement is transposed: the proof can be made by means of the same terms.
Also if the affirmative premiss is partially false, the negative wholly
true, a true conclusion is possible. For nothing prevents A belonging to
some B, but not to C as a whole, while B belongs to no C, e.g. animal belongs
to some white things, but to no pitch, and white belongs to no pitch. Consequently
if it is assumed that A belongs to the whole of B, but to no C, the premiss
AB is partially false, the premiss AC is wholly true, and the conclusion
is true.

(4) And if both the premisses are partially false, the conclusion
may be true. For it is possible that A should belong to some B and to some
C, and B to no C, e.g. animal to some white things and to some black things,
though white belongs to nothing black. If then it is assumed that A belongs
to all B and to no C, both premisses are partially false, but the conclusion
is true. Similarly, if the negative premiss is transposed, the proof can
be made by means of the same terms.

It is clear also that our thesis holds in particular syllogisms.
For (5) nothing prevents A belonging to all B and to some C, though B does
not belong to some C, e.g. animal to every man and to some white things,
though man will not belong to some white things. If then it is stated that
A belongs to no B and to some C, the universal premiss is wholly false,
the particular premiss is true, and the conclusion is true. Similarly if
the premiss AB is affirmative: for it is possible that A should belong
to no B, and not to some C, though B does not belong to some C, e.g. animal
belongs to nothing lifeless, and does not belong to some white things,
and lifeless will not belong to some white things. If then it is stated
that A belongs to all B and not to some C, the premiss AB which is universal
is wholly false, the premiss AC is true, and the conclusion is true. Also
a true conclusion is possible when the universal premiss is true, and the
particular is false. For nothing prevents A following neither B nor C at
all, while B does not belong to some C, e.g. animal belongs to no number
nor to anything lifeless, and number does not follow some lifeless things.
If then it is stated that A belongs to no B and to some C, the conclusion
will be true, and the universal premiss true, but the particular false.
Similarly if the premiss which is stated universally is affirmative. For
it is possible that should A belong both to B and to C as wholes, though
B does not follow some C, e.g. a genus in relation to its species and difference:
for animal follows every man and footed things as a whole, but man does
not follow every footed thing. Consequently if it is assumed that A belongs
to the whole of B, but does not belong to some C, the universal premiss
is true, the particular false, and the conclusion true.

(6) It is clear too that though both premisses are false they may
yield a true conclusion, since it is possible that A should belong both
to B and to C as wholes, though B does not follow some C. For if it is
assumed that A belongs to no B and to some C, the premisses are both false,
but the conclusion is true. Similarly if the universal premiss is affirmative
and the particular negative. For it is possible that A should follow no
B and all C, though B does not belong to some C, e.g. animal follows no
science but every man, though science does not follow every man. If then
A is assumed to belong to the whole of B, and not to follow some C, the
premisses are false but the conclusion is true.

Part 4

In the last figure a true conclusion may come through what is false,
alike when both premisses are wholly false, when each is partly false,
when one premiss is wholly true, the other false, when one premiss is partly
false, the other wholly true, and vice versa, and in every other way in
which it is possible to alter the premisses. For (1) nothing prevents neither
A nor B from belonging to any C, while A belongs to some B, e.g. neither
man nor footed follows anything lifeless, though man belongs to some footed
things. If then it is assumed that A and B belong to all C, the premisses
will be wholly false, but the conclusion true. Similarly if one premiss
is negative, the other affirmative. For it is possible that B should belong
to no C, but A to all C, and that should not belong to some B, e.g. black
belongs to no swan, animal to every swan, and animal not to everything
black. Consequently if it is assumed that B belongs to all C, and A to
no C, A will not belong to some B: and the conclusion is true, though the
premisses are false.

(2) Also if each premiss is partly false, the conclusion may be
true. For nothing prevents both A and B from belonging to some C while
A belongs to some B, e.g. white and beautiful belong to some animals, and
white to some beautiful things. If then it is stated that A and B belong
to all C, the premisses are partially false, but the conclusion is true.
Similarly if the premiss AC is stated as negative. For nothing prevents
A from not belonging, and B from belonging, to some C, while A does not
belong to all B, e.g. white does not belong to some animals, beautiful
belongs to some animals, and white does not belong to everything beautiful.
Consequently if it is assumed that A belongs to no C, and B to all C, both
premisses are partly false, but the conclusion is true.

(3) Similarly if one of the premisses assumed is wholly false,
the other wholly true. For it is possible that both A and B should follow
all C, though A does not belong to some B, e.g. animal and white follow
every swan, though animal does not belong to everything white. Taking these
then as terms, if one assumes that B belongs to the whole of C, but A does
not belong to C at all, the premiss BC will be wholly true, the premiss
AC wholly false, and the conclusion true. Similarly if the statement BC
is false, the statement AC true, the conclusion may be true. The same terms
will serve for the proof. Also if both the premisses assumed are affirmative,
the conclusion may be true. For nothing prevents B from following all C,
and A from not belonging to C at all, though A belongs to some B, e.g.
animal belongs to every swan, black to no swan, and black to some animals.
Consequently if it is assumed that A and B belong to every C, the premiss
BC is wholly true, the premiss AC is wholly false, and the conclusion is
true. Similarly if the premiss AC which is assumed is true: the proof can
be made through the same terms.

(4) Again if one premiss is wholly true, the other partly false,
the conclusion may be true. For it is possible that B should belong to
all C, and A to some C, while A belongs to some B, e.g. biped belongs to
every man, beautiful not to every man, and beautiful to some bipeds. If
then it is assumed that both A and B belong to the whole of C, the premiss
BC is wholly true, the premiss AC partly false, the conclusion true. Similarly
if of the premisses assumed AC is true and BC partly false, a true conclusion
is possible: this can be proved, if the same terms as before are transposed.
Also the conclusion may be true if one premiss is negative, the other affirmative.
For since it is possible that B should belong to the whole of C, and A
to some C, and, when they are so, that A should not belong to all B, therefore
it is assumed that B belongs to the whole of C, and A to no C, the negative
premiss is partly false, the other premiss wholly true, and the conclusion
is true. Again since it has been proved that if A belongs to no C and B
to some C, it is possible that A should not belong to some C, it is clear
that if the premiss AC is wholly true, and the premiss BC partly false,
it is possible that the conclusion should be true. For if it is assumed
that A belongs to no C, and B to all C, the premiss AC is wholly true,
and the premiss BC is partly false.

(5) It is clear also in the case of particular syllogisms that
a true conclusion may come through what is false, in every possible way.
For the same terms must be taken as have been taken when the premisses
are universal, positive terms in positive syllogisms, negative terms in
negative. For it makes no difference to the setting out of the terms, whether
one assumes that what belongs to none belongs to all or that what belongs
to some belongs to all. The same applies to negative
statements.

It is clear then that if the conclusion is false, the premisses
of the argument must be false, either all or some of them; but when the
conclusion is true, it is not necessary that the premisses should be true,
either one or all, yet it is possible, though no part of the syllogism
is true, that the conclusion may none the less be true; but it is not necessitated.
The reason is that when two things are so related to one another, that
if the one is, the other necessarily is, then if the latter is not, the
former will not be either, but if the latter is, it is not necessary that
the former should be. But it is impossible that the same thing should be
necessitated by the being and by the not-being of the same thing. I mean,
for example, that it is impossible that B should necessarily be great since
A is white and that B should necessarily be great since A is not white.
For whenever since this, A, is white it is necessary that that, B, should
be great, and since B is great that C should not be white, then it is necessary
if is white that C should not be white. And whenever it is necessary, since
one of two things is, that the other should be, it is necessary, if the
latter is not, that the former (viz. A) should not be. If then B is not
great A cannot be white. But if, when A is not white, it is necessary that
B should be great, it necessarily results that if B is not great, B itself
is great. (But this is impossible.) For if B is not great, A will necessarily
not be white. If then when this is not white B must be great, it results
that if B is not great, it is great, just as if it were proved through
three terms.

Part 5

Circular and reciprocal proof means proof by means of the conclusion,
i.e. by converting one of the premisses simply and inferring the premiss
which was assumed in the original syllogism: e.g. suppose it has been necessary
to prove that A belongs to all C, and it has been proved through B; suppose
that A should now be proved to belong to B by assuming that A belongs to
C, and C to B-so A belongs to B: but in the first syllogism the converse
was assumed, viz. that B belongs to C. Or suppose it is necessary to prove
that B belongs to C, and A is assumed to belong to C, which was the conclusion
of the first syllogism, and B to belong to A but the converse was assumed
in the earlier syllogism, viz. that A belongs to B. In no other way is
reciprocal proof possible. If another term is taken as middle, the proof
is not circular: for neither of the propositions assumed is the same as
before: if one of the accepted terms is taken as middle, only one of the
premisses of the first syllogism can be assumed in the second: for if both
of them are taken the same conclusion as before will result: but it must
be different. If the terms are not convertible, one of the premisses from
which the syllogism results must be undemonstrated: for it is not possible
to demonstrate through these terms that the third belongs to the middle
or the middle to the first. If the terms are convertible, it is possible
to demonstrate everything reciprocally, e.g. if A and B and C are convertible
with one another. Suppose the proposition AC has been demonstrated through
B as middle term, and again the proposition AB through the conclusion and
the premiss BC converted, and similarly the proposition BC through the
conclusion and the premiss AB converted. But it is necessary to prove both
the premiss CB, and the premiss BA: for we have used these alone without
demonstrating them. If then it is assumed that B belongs to all C, and
C to all A, we shall have a syllogism relating B to A. Again if it is assumed
that C belongs to all A, and A to all B, C must belong to all B. In both
these syllogisms the premiss CA has been assumed without being demonstrated:
the other premisses had ex hypothesi been proved. Consequently if we succeed
in demonstrating this premiss, all the premisses will have been proved
reciprocally. If then it is assumed that C belongs to all B, and B to all
A, both the premisses assumed have been proved, and C must belong to A.
It is clear then that only if the terms are convertible is circular and
reciprocal demonstration possible (if the terms are not convertible, the
matter stands as we said above). But it turns out in these also that we
use for the demonstration the very thing that is being proved: for C is
proved of B, and B of by assuming that C is said of and C is proved of
A through these premisses, so that we use the conclusion for the
demonstration.

In negative syllogisms reciprocal proof is as follows. Let B belong
to all C, and A to none of the Bs: we conclude that A belongs to none of
the Cs. If again it is necessary to prove that A belongs to none of the
Bs (which was previously assumed) A must belong to no C, and C to all B:
thus the previous premiss is reversed. If it is necessary to prove that
B belongs to C, the proposition AB must no longer be converted as before:
for the premiss 'B belongs to no A' is identical with the premiss 'A belongs
to no B'. But we must assume that B belongs to all of that to none of which
longs. Let A belong to none of the Cs (which was the previous conclusion)
and assume that B belongs to all of that to none of which A belongs. It
is necessary then that B should belong to all C. Consequently each of the
three propositions has been made a conclusion, and this is circular demonstration,
to assume the conclusion and the converse of one of the premisses, and
deduce the remaining premiss.

In particular syllogisms it is not possible to demonstrate the
universal premiss through the other propositions, but the particular premiss
can be demonstrated. Clearly it is impossible to demonstrate the universal
premiss: for what is universal is proved through propositions which are
universal, but the conclusion is not universal, and the proof must start
from the conclusion and the other premiss. Further a syllogism cannot be
made at all if the other premiss is converted: for the result is that both
premisses are particular. But the particular premiss may be proved. Suppose
that A has been proved of some C through B. If then it is assumed that
B belongs to all A and the conclusion is retained, B will belong to some
C: for we obtain the first figure and A is middle. But if the syllogism
is negative, it is not possible to prove the universal premiss, for the
reason given above. But it is possible to prove the particular premiss,
if the proposition AB is converted as in the universal syllogism, i.e 'B
belongs to some of that to some of which A does not belong': otherwise
no syllogism results because the particular premiss is
negative.

Part 6

In the second figure it is not possible to prove an affirmative
proposition in this way, but a negative proposition may be proved. An affirmative
proposition is not proved because both premisses of the new syllogism are
not affirmative (for the conclusion is negative) but an affirmative proposition
is (as we saw) proved from premisses which are both affirmative. The negative
is proved as follows. Let A belong to all B, and to no C: we conclude that
B belongs to no C. If then it is assumed that B belongs to all A, it is
necessary that A should belong to no C: for we get the second figure, with
B as middle. But if the premiss AB was negative, and the other affirmative,
we shall have the first figure. For C belongs to all A and B to no C, consequently
B belongs to no A: neither then does A belong to B. Through the conclusion,
therefore, and one premiss, we get no syllogism, but if another premiss
is assumed in addition, a syllogism will be possible. But if the syllogism
not universal, the universal premiss cannot be proved, for the same reason
as we gave above, but the particular premiss can be proved whenever the
universal statement is affirmative. Let A belong to all B, and not to all
C: the conclusion is BC. If then it is assumed that B belongs to all A,
but not to all C, A will not belong to some C, B being middle. But if the
universal premiss is negative, the premiss AC will not be demonstrated
by the conversion of AB: for it turns out that either both or one of the
premisses is negative; consequently a syllogism will not be possible. But
the proof will proceed as in the universal syllogisms, if it is assumed
that A belongs to some of that to some of which B does not
belong.

Part 7

In the third figure, when both premisses are taken universally,
it is not possible to prove them reciprocally: for that which is universal
is proved through statements which are universal, but the conclusion in
this figure is always particular, so that it is clear that it is not possible
at all to prove through this figure the universal premiss. But if one premiss
is universal, the other particular, proof of the latter will sometimes
be possible, sometimes not. When both the premisses assumed are affirmative,
and the universal concerns the minor extreme, proof will be possible, but
when it concerns the other extreme, impossible. Let A belong to all C and
B to some C: the conclusion is the statement AB. If then it is assumed
that C belongs to all A, it has been proved that C belongs to some B, but
that B belongs to some C has not been proved. And yet it is necessary,
if C belongs to some B, that B should belong to some C. But it is not the
same that this should belong to that, and that to this: but we must assume
besides that if this belongs to some of that, that belongs to some of this.
But if this is assumed the syllogism no longer results from the conclusion
and the other premiss. But if B belongs to all C, and A to some C, it will
be possible to prove the proposition AC, when it is assumed that C belongs
to all B, and A to some B. For if C belongs to all B and A to some B, it
is necessary that A should belong to some C, B being middle. And whenever
one premiss is affirmative the other negative, and the affirmative is universal,
the other premiss can be proved. Let B belong to all C, and A not to some
C: the conclusion is that A does not belong to some B. If then it is assumed
further that C belongs to all B, it is necessary that A should not belong
to some C, B being middle. But when the negative premiss is universal,
the other premiss is not except as before, viz. if it is assumed that that
belongs to some of that, to some of which this does not belong, e.g. if
A belongs to no C, and B to some C: the conclusion is that A does not belong
to some B. If then it is assumed that C belongs to some of that to some
of which does not belong, it is necessary that C should belong to some
of the Bs. In no other way is it possible by converting the universal premiss
to prove the other: for in no other way can a syllogism be
formed.

It is clear then that in the first figure reciprocal proof is made
both through the third and through the first figure-if the conclusion is
affirmative through the first; if the conclusion is negative through the
last. For it is assumed that that belongs to all of that to none of which
this belongs. In the middle figure, when the syllogism is universal, proof
is possible through the second figure and through the first, but when particular
through the second and the last. In the third figure all proofs are made
through itself. It is clear also that in the third figure and in the middle
figure those syllogisms which are not made through those figures themselves
either are not of the nature of circular proof or are
imperfect.

Part 8

To convert a syllogism means to alter the conclusion and make another
syllogism to prove that either the extreme cannot belong to the middle
or the middle to the last term. For it is necessary, if the conclusion
has been changed into its opposite and one of the premisses stands, that
the other premiss should be destroyed. For if it should stand, the conclusion
also must stand. It makes a difference whether the conclusion is converted
into its contradictory or into its contrary. For the same syllogism does
not result whichever form the conversion takes. This will be made clear
by the sequel. By contradictory opposition I mean the opposition of 'to
all' to 'not to all', and of 'to some' to 'to none'; by contrary opposition
I mean the opposition of 'to all' to 'to none', and of 'to some' to 'not
to some'. Suppose that A been proved of C, through B as middle term. If
then it should be assumed that A belongs to no C, but to all B, B will
belong to no C. And if A belongs to no C, and B to all C, A will belong,
not to no B at all, but not to all B. For (as we saw) the universal is
not proved through the last figure. In a word it is not possible to refute
universally by conversion the premiss which concerns the major extreme:
for the refutation always proceeds through the third since it is necessary
to take both premisses in reference to the minor extreme. Similarly if
the syllogism is negative. Suppose it has been proved that A belongs to
no C through B. Then if it is assumed that A belongs to all C, and to no
B, B will belong to none of the Cs. And if A and B belong to all C, A will
belong to some B: but in the original premiss it belonged to no
B.

If the conclusion is converted into its contradictory, the syllogisms
will be contradictory and not universal. For one premiss is particular,
so that the conclusion also will be particular. Let the syllogism be affirmative,
and let it be converted as stated. Then if A belongs not to all C, but
to all B, B will belong not to all C. And if A belongs not to all C, but
B belongs to all C, A will belong not to all B. Similarly if the syllogism
is negative. For if A belongs to some C, and to no B, B will belong, not
to no C at all, but-not to some C. And if A belongs to some C, and B to
all C, as was originally assumed, A will belong to some
B.

In particular syllogisms when the conclusion is converted into
its contradictory, both premisses may be refuted, but when it is converted
into its contrary, neither. For the result is no longer, as in the universal
syllogisms, refutation in which the conclusion reached by O, conversion
lacks universality, but no refutation at all. Suppose that A has been proved
of some C. If then it is assumed that A belongs to no C, and B to some
C, A will not belong to some B: and if A belongs to no C, but to all B,
B will belong to no C. Thus both premisses are refuted. But neither can
be refuted if the conclusion is converted into its contrary. For if A does
not belong to some C, but to all B, then B will not belong to some C. But
the original premiss is not yet refuted: for it is possible that B should
belong to some C, and should not belong to some C. The universal premiss
AB cannot be affected by a syllogism at all: for if A does not belong to
some of the Cs, but B belongs to some of the Cs, neither of the premisses
is universal. Similarly if the syllogism is negative: for if it should
be assumed that A belongs to all C, both premisses are refuted: but if
the assumption is that A belongs to some C, neither premiss is refuted.
The proof is the same as before.

Part 9

In the second figure it is not possible to refute the premiss which
concerns the major extreme by establishing something contrary to it, whichever
form the conversion of the conclusion may take. For the conclusion of the
refutation will always be in the third figure, and in this figure (as we
saw) there is no universal syllogism. The other premiss can be refuted
in a manner similar to the conversion: I mean, if the conclusion of the
first syllogism is converted into its contrary, the conclusion of the refutation
will be the contrary of the minor premiss of the first, if into its contradictory,
the contradictory. Let A belong to all B and to no C: conclusion BC. If
then it is assumed that B belongs to all C, and the proposition AB stands,
A will belong to all C, since the first figure is produced. If B belongs
to all C, and A to no C, then A belongs not to all B: the figure is the
last. But if the conclusion BC is converted into its contradictory, the
premiss AB will be refuted as before, the premiss, AC by its contradictory.
For if B belongs to some C, and A to no C, then A will not belong to some
B. Again if B belongs to some C, and A to all B, A will belong to some
C, so that the syllogism results in the contradictory of the minor premiss.
A similar proof can be given if the premisses are transposed in respect
of their quality.

If the syllogism is particular, when the conclusion is converted
into its contrary neither premiss can be refuted, as also happened in the
first figure,' if the conclusion is converted into its contradictory, both
premisses can be refuted. Suppose that A belongs to no B, and to some C:
the conclusion is BC. If then it is assumed that B belongs to some C, and
the statement AB stands, the conclusion will be that A does not belong
to some C. But the original statement has not been refuted: for it is possible
that A should belong to some C and also not to some C. Again if B belongs
to some C and A to some C, no syllogism will be possible: for neither of
the premisses taken is universal. Consequently the proposition AB is not
refuted. But if the conclusion is converted into its contradictory, both
premisses can be refuted. For if B belongs to all C, and A to no B, A will
belong to no C: but it was assumed to belong to some C. Again if B belongs
to all C and A to some C, A will belong to some B. The same proof can be
given if the universal statement is affirmative.

Part 10

In the third figure when the conclusion is converted into its contrary,
neither of the premisses can be refuted in any of the syllogisms, but when
the conclusion is converted into its contradictory, both premisses may
be refuted and in all the moods. Suppose it has been proved that A belongs
to some B, C being taken as middle, and the premisses being universal.
If then it is assumed that A does not belong to some B, but B belongs to
all C, no syllogism is formed about A and C. Nor if A does not belong to
some B, but belongs to all C, will a syllogism be possible about B and
C. A similar proof can be given if the premisses are not universal. For
either both premisses arrived at by the conversion must be particular,
or the universal premiss must refer to the minor extreme. But we found
that no syllogism is possible thus either in the first or in the middle
figure. But if the conclusion is converted into its contradictory, both
the premisses can be refuted. For if A belongs to no B, and B to all C,
then A belongs to no C: again if A belongs to no B, and to all C, B belongs
to no C. And similarly if one of the premisses is not universal. For if
A belongs to no B, and B to some C, A will not belong to some C: if A belongs
to no B, and to C, B will belong to no C.

Similarly if the original syllogism is negative. Suppose it has
been proved that A does not belong to some B, BC being affirmative, AC
being negative: for it was thus that, as we saw, a syllogism could be made.
Whenever then the contrary of the conclusion is assumed a syllogism will
not be possible. For if A belongs to some B, and B to all C, no syllogism
is possible (as we saw) about A and C. Nor, if A belongs to some B, and
to no C, was a syllogism possible concerning B and C. Therefore the premisses
are not refuted. But when the contradictory of the conclusion is assumed,
they are refuted. For if A belongs to all B, and B to C, A belongs to all
C: but A was supposed originally to belong to no C. Again if A belongs
to all B, and to no C, then B belongs to no C: but it was supposed to belong
to all C. A similar proof is possible if the premisses are not universal.
For AC becomes universal and negative, the other premiss particular and
affirmative. If then A belongs to all B, and B to some C, it results that
A belongs to some C: but it was supposed to belong to no C. Again if A
belongs to all B, and to no C, then B belongs to no C: but it was assumed
to belong to some C. If A belongs to some B and B to some C, no syllogism
results: nor yet if A belongs to some B, and to no C. Thus in one way the
premisses are refuted, in the other way they are not.

From what has been said it is clear how a syllogism results in
each figure when the conclusion is converted; when a result contrary to
the premiss, and when a result contradictory to the premiss, is obtained.
It is clear that in the first figure the syllogisms are formed through
the middle and the last figures, and the premiss which concerns the minor
extreme is alway refuted through the middle figure, the premiss which concerns
the major through the last figure. In the second figure syllogisms proceed
through the first and the last figures, and the premiss which concerns
the minor extreme is always refuted through the first figure, the premiss
which concerns the major extreme through the last. In the third figure
the refutation proceeds through the first and the middle figures; the premiss
which concerns the major is always refuted through the first figure, the
premiss which concerns the minor through the middle
figure.

Part 11

It is clear then what conversion is, how it is effected in each
figure, and what syllogism results. The syllogism per impossibile is proved
when the contradictory of the conclusion stated and another premiss is
assumed; it can be made in all the figures. For it resembles conversion,
differing only in this: conversion takes place after a syllogism has been
formed and both the premisses have been taken, but a reduction to the impossible
takes place not because the contradictory has been agreed to already, but
because it is clear that it is true. The terms are alike in both, and the
premisses of both are taken in the same way. For example if A belongs to
all B, C being middle, then if it is supposed that A does not belong to
all B or belongs to no B, but to all C (which was admitted to be true),
it follows that C belongs to no B or not to all B. But this is impossible:
consequently the supposition is false: its contradictory then is true.
Similarly in the other figures: for whatever moods admit of conversion
admit also of the reduction per impossibile.

All the problems can be proved per impossibile in all the figures,
excepting the universal affirmative, which is proved in the middle and
third figures, but not in the first. Suppose that A belongs not to all
B, or to no B, and take besides another premiss concerning either of the
terms, viz. that C belongs to all A, or that B belongs to all D; thus we
get the first figure. If then it is supposed that A does not belong to
all B, no syllogism results whichever term the assumed premiss concerns;
but if it is supposed that A belongs to no B, when the premiss BD is assumed
as well we shall prove syllogistically what is false, but not the problem
proposed. For if A belongs to no B, and B belongs to all D, A belongs to
no D. Let this be impossible: it is false then A belongs to no B. But the
universal affirmative is not necessarily true if the universal negative
is false. But if the premiss CA is assumed as well, no syllogism results,
nor does it do so when it is supposed that A does not belong to all B.
Consequently it is clear that the universal affirmative cannot be proved
in the first figure per impossibile.

But the particular affirmative and the universal and particular
negatives can all be proved. Suppose that A belongs to no B, and let it
have been assumed that B belongs to all or to some C. Then it is necessary
that A should belong to no C or not to all C. But this is impossible (for
let it be true and clear that A belongs to all C): consequently if this
is false, it is necessary that A should belong to some B. But if the other
premiss assumed relates to A, no syllogism will be possible. Nor can a
conclusion be drawn when the contrary of the conclusion is supposed, e.g.
that A does not belong to some B. Clearly then we must suppose the
contradictory.

Again suppose that A belongs to some B, and let it have been assumed
that C belongs to all A. It is necessary then that C should belong to some
B. But let this be impossible, so that the supposition is false: in that
case it is true that A belongs to no B. We may proceed in the same way
if the proposition CA has been taken as negative. But if the premiss assumed
concerns B, no syllogism will be possible. If the contrary is supposed,
we shall have a syllogism and an impossible conclusion, but the problem
in hand is not proved. Suppose that A belongs to all B, and let it have
been assumed that C belongs to all A. It is necessary then that C should
belong to all B. But this is impossible, so that it is false that A belongs
to all B. But we have not yet shown it to be necessary that A belongs to
no B, if it does not belong to all B. Similarly if the other premiss taken
concerns B; we shall have a syllogism and a conclusion which is impossible,
but the hypothesis is not refuted. Therefore it is the contradictory that
we must suppose.

To prove that A does not belong to all B, we must suppose that
it belongs to all B: for if A belongs to all B, and C to all A, then C
belongs to all B; so that if this is impossible, the hypothesis is false.
Similarly if the other premiss assumed concerns B. The same results if
the original proposition CA was negative: for thus also we get a syllogism.
But if the negative proposition concerns B, nothing is proved. If the hypothesis
is that A belongs not to all but to some B, it is not proved that A belongs
not to all B, but that it belongs to no B. For if A belongs to some B,
and C to all A, then C will belong to some B. If then this is impossible,
it is false that A belongs to some B; consequently it is true that A belongs
to no B. But if this is proved, the truth is refuted as well; for the original
conclusion was that A belongs to some B, and does not belong to some B.
Further the impossible does not result from the hypothesis: for then the
hypothesis would be false, since it is impossible to draw a false conclusion
from true premisses: but in fact it is true: for A belongs to some B. Consequently
we must not suppose that A belongs to some B, but that it belongs to all
B. Similarly if we should be proving that A does not belong to some B:
for if 'not to belong to some' and 'to belong not to all' have the same
meaning, the demonstration of both will be identical.

It is clear then that not the contrary but the contradictory ought
to be supposed in all the syllogisms. For thus we shall have necessity
of inference, and the claim we make is one that will be generally accepted.
For if of everything one or other of two contradictory statements holds
good, then if it is proved that the negation does not hold, the affirmation
must be true. Again if it is not admitted that the affirmation is true,
the claim that the negation is true will be generally accepted. But in
neither way does it suit to maintain the contrary: for it is not necessary
that if the universal negative is false, the universal affirmative should
be true, nor is it generally accepted that if the one is false the other
is true.

Part 12

It is clear then that in the first figure all problems except the
universal affirmative are proved per impossibile. But in the middle and
the last figures this also is proved. Suppose that A does not belong to
all B, and let it have been assumed that A belongs to all C. If then A
belongs not to all B, but to all C, C will not belong to all B. But this
is impossible (for suppose it to be clear that C belongs to all B): consequently
the hypothesis is false. It is true then that A belongs to all B. But if
the contrary is supposed, we shall have a syllogism and a result which
is impossible: but the problem in hand is not proved. For if A belongs
to no B, and to all C, C will belong to no B. This is impossible; so that
it is false that A belongs to no B. But though this is false, it does not
follow that it is true that A belongs to all B.

When A belongs to some B, suppose that A belongs to no B, and let
A belong to all C. It is necessary then that C should belong to no B. Consequently,
if this is impossible, A must belong to some B. But if it is supposed that
A does not belong to some B, we shall have the same results as in the first
figure.

Again suppose that A belongs to some B, and let A belong to no
C. It is necessary then that C should not belong to some B. But originally
it belonged to all B, consequently the hypothesis is false: A then will
belong to no B.

When A does not belong to an B, suppose it does belong to all B,
and to no C. It is necessary then that C should belong to no B. But this
is impossible: so that it is true that A does not belong to all B. It is
clear then that all the syllogisms can be formed in the middle
figure.

Part 13

Similarly they can all be formed in the last figure. Suppose that
A does not belong to some B, but C belongs to all B: then A does not belong
to some C. If then this is impossible, it is false that A does not belong
to some B; so that it is true that A belongs to all B. But if it is supposed
that A belongs to no B, we shall have a syllogism and a conclusion which
is impossible: but the problem in hand is not proved: for if the contrary
is supposed, we shall have the same results as before.

But to prove that A belongs to some B, this hypothesis must be
made. If A belongs to no B, and C to some B, A will belong not to all C.
If then this is false, it is true that A belongs to some
B.

When A belongs to no B, suppose A belongs to some B, and let it
have been assumed that C belongs to all B. Then it is necessary that A
should belong to some C. But ex hypothesi it belongs to no C, so that it
is false that A belongs to some B. But if it is supposed that A belongs
to all B, the problem is not proved.

But this hypothesis must be made if we are prove that A belongs
not to all B. For if A belongs to all B and C to some B, then A belongs
to some C. But this we assumed not to be so, so it is false that A belongs
to all B. But in that case it is true that A belongs not to all B. If however
it is assumed that A belongs to some B, we shall have the same result as
before.

It is clear then that in all the syllogisms which proceed per impossibile
the contradictory must be assumed. And it is plain that in the middle figure
an affirmative conclusion, and in the last figure a universal conclusion,
are proved in a way.

Part 14

Demonstration per impossibile differs from ostensive proof in that
it posits what it wishes to refute by reduction to a statement admitted
to be false; whereas ostensive proof starts from admitted positions. Both,
indeed, take two premisses that are admitted, but the latter takes the
premisses from which the syllogism starts, the former takes one of these,
along with the contradictory of the original conclusion. Also in the ostensive
proof it is not necessary that the conclusion should be known, nor that
one should suppose beforehand that it is true or not: in the other it is
necessary to suppose beforehand that it is not true. It makes no difference
whether the conclusion is affirmative or negative; the method is the same
in both cases. Everything which is concluded ostensively can be proved
per impossibile, and that which is proved per impossibile can be proved
ostensively, through the same terms. Whenever the syllogism is formed in
the first figure, the truth will be found in the middle or the last figure,
if negative in the middle, if affirmative in the last. Whenever the syllogism
is formed in the middle figure, the truth will be found in the first, whatever
the problem may be. Whenever the syllogism is formed in the last figure,
the truth will be found in the first and middle figures, if affirmative
in first, if negative in the middle. Suppose that A has been proved to
belong to no B, or not to all B, through the first figure. Then the hypothesis
must have been that A belongs to some B, and the original premisses that
C belongs to all A and to no B. For thus the syllogism was made and the
impossible conclusion reached. But this is the middle figure, if C belongs
to all A and to no B. And it is clear from these premisses that A belongs
to no B. Similarly if has been proved not to belong to all B. For the hypothesis
is that A belongs to all B; and the original premisses are that C belongs
to all A but not to all B. Similarly too, if the premiss CA should be negative:
for thus also we have the middle figure. Again suppose it has been proved
that A belongs to some B. The hypothesis here is that is that A belongs
to no B; and the original premisses that B belongs to all C, and A either
to all or to some C: for in this way we shall get what is impossible. But
if A and B belong to all C, we have the last figure. And it is clear from
these premisses that A must belong to some B. Similarly if B or A should
be assumed to belong to some C.

Again suppose it has been proved in the middle figure that A belongs
to all B. Then the hypothesis must have been that A belongs not to all
B, and the original premisses that A belongs to all C, and C to all B:
for thus we shall get what is impossible. But if A belongs to all C, and
C to all B, we have the first figure. Similarly if it has been proved that
A belongs to some B: for the hypothesis then must have been that A belongs
to no B, and the original premisses that A belongs to all C, and C to some
B. If the syllogism is negative, the hypothesis must have been that A belongs
to some B, and the original premisses that A belongs to no C, and C to
all B, so that the first figure results. If the syllogism is not universal,
but proof has been given that A does not belong to some B, we may infer
in the same way. The hypothesis is that A belongs to all B, the original
premisses that A belongs to no C, and C belongs to some B: for thus we
get the first figure.

Again suppose it has been proved in the third figure that A belongs
to all B. Then the hypothesis must have been that A belongs not to all
B, and the original premisses that C belongs to all B, and A belongs to
all C; for thus we shall get what is impossible. And the original premisses
form the first figure. Similarly if the demonstration establishes a particular
proposition: the hypothesis then must have been that A belongs to no B,
and the original premisses that C belongs to some B, and A to all C. If
the syllogism is negative, the hypothesis must have been that A belongs
to some B, and the original premisses that C belongs to no A and to all
B, and this is the middle figure. Similarly if the demonstration is not
universal. The hypothesis will then be that A belongs to all B, the premisses
that C belongs to no A and to some B: and this is the middle
figure.

It is clear then that it is possible through the same terms to
prove each of the problems ostensively as well. Similarly it will be possible
if the syllogisms are ostensive to reduce them ad impossibile in the terms
which have been taken, whenever the contradictory of the conclusion of
the ostensive syllogism is taken as a premiss. For the syllogisms become
identical with those which are obtained by means of conversion, so that
we obtain immediately the figures through which each problem will be solved.
It is clear then that every thesis can be proved in both ways, i.e. per
impossibile and ostensively, and it is not possible to separate one method
from the other.

Part 15

In what figure it is possible to draw a conclusion from premisses
which are opposed, and in what figure this is not possible, will be made
clear in this way. Verbally four kinds of opposition are possible, viz.
universal affirmative to universal negative, universal affirmative to particular
negative, particular affirmative to universal negative, and particular
affirmative to particular negative: but really there are only three: for
the particular affirmative is only verbally opposed to the particular negative.
Of the genuine opposites I call those which are universal contraries, the
universal affirmative and the universal negative, e.g. 'every science is
good', 'no science is good'; the others I call contradictories.

In the first figure no syllogism whether affirmative or negative
can be made out of opposed premisses: no affirmative syllogism is possible
because both premisses must be affirmative, but opposites are, the one
affirmative, the other negative: no negative syllogism is possible because
opposites affirm and deny the same predicate of the same subject, and the
middle term in the first figure is not predicated of both extremes, but
one thing is denied of it, and it is affirmed of something else: but such
premisses are not opposed.

In the middle figure a syllogism can be made both oLcontradictories
and of contraries. Let A stand for good, let B and C stand for science.
If then one assumes that every science is good, and no science is good,
A belongs to all B and to no C, so that B belongs to no C: no science then
is a science. Similarly if after taking 'every science is good' one took
'the science of medicine is not good'; for A belongs to all B but to no
C, so that a particular science will not be a science. Again, a particular
science will not be a science if A belongs to all C but to no B, and B
is science, C medicine, and A supposition: for after taking 'no science
is supposition', one has assumed that a particular science is supposition.
This syllogism differs from the preceding because the relations between
the terms are reversed: before, the affirmative statement concerned B,
now it concerns C. Similarly if one premiss is not universal: for the middle
term is always that which is stated negatively of one extreme, and affirmatively
of the other. Consequently it is possible that contradictories may lead
to a conclusion, though not always or in every mood, but only if the terms
subordinate to the middle are such that they are either identical or related
as whole to part. Otherwise it is impossible: for the premisses cannot
anyhow be either contraries or contradictories.

In the third figure an affirmative syllogism can never be made
out of opposite premisses, for the reason given in reference to the first
figure; but a negative syllogism is possible whether the terms are universal
or not. Let B and C stand for science, A for medicine. If then one should
assume that all medicine is science and that no medicine is science, he
has assumed that B belongs to all A and C to no A, so that a particular
science will not be a science. Similarly if the premiss BA is not assumed
universally. For if some medicine is science and again no medicine is science,
it results that some science is not science, The premisses are contrary
if the terms are taken universally; if one is particular, they are
contradictory.

We must recognize that it is possible to take opposites in the
way we said, viz. 'all science is good' and 'no science is good' or 'some
science is not good'. This does not usually escape notice. But it is possible
to establish one part of a contradiction through other premisses, or to
assume it in the way suggested in the Topics. Since there are three oppositions
to affirmative statements, it follows that opposite statements may be assumed
as premisses in six ways; we may have either universal affirmative and
negative, or universal affirmative and particular negative, or particular
affirmative and universal negative, and the relations between the terms
may be reversed; e.g. A may belong to all B and to no C, or to all C and
to no B, or to all of the one, not to all of the other; here too the relation
between the terms may be reversed. Similarly in the third figure. So it
is clear in how many ways and in what figures a syllogism can be made by
means of premisses which are opposed.

It is clear too that from false premisses it is possible to draw
a true conclusion, as has been said before, but it is not possible if the
premisses are opposed. For the syllogism is always contrary to the fact,
e.g. if a thing is good, it is proved that it is not good, if an animal,
that it is not an animal because the syllogism springs out of a contradiction
and the terms presupposed are either identical or related as whole and
part. It is evident also that in fallacious reasonings nothing prevents
a contradiction to the hypothesis from resulting, e.g. if something is
odd, it is not odd. For the syllogism owed its contrariety to its contradictory
premisses; if we assume such premisses we shall get a result that contradicts
our hypothesis. But we must recognize that contraries cannot be inferred
from a single syllogism in such a way that we conclude that what is not
good is good, or anything of that sort unless a self-contradictory premiss
is at once assumed, e.g. 'every animal is white and not white', and we
proceed 'man is an animal'. Either we must introduce the contradiction
by an additional assumption, assuming, e.g., that every science is supposition,
and then assuming 'Medicine is a science, but none of it is supposition'
(which is the mode in which refutations are made), or we must argue from
two syllogisms. In no other way than this, as was said before, is it possible
that the premisses should be really contrary.

Part 16

To beg and assume the original question is a species of failure
to demonstrate the problem proposed; but this happens in many ways. A man
may not reason syllogistically at all, or he may argue from premisses which
are less known or equally unknown, or he may establish the antecedent by
means of its consequents; for demonstration proceeds from what is more
certain and is prior. Now begging the question is none of these: but since
we get to know some things naturally through themselves, and other things
by means of something else (the first principles through themselves, what
is subordinate to them through something else), whenever a man tries to
prove what is not self-evident by means of itself, then he begs the original
question. This may be done by assuming what is in question at once; it
is also possible to make a transition to other things which would naturally
be proved through the thesis proposed, and demonstrate it through them,
e.g. if A should be proved through B, and B through C, though it was natural
that C should be proved through A: for it turns out that those who reason
thus are proving A by means of itself. This is what those persons do who
suppose that they are constructing parallel straight lines: for they fail
to see that they are assuming facts which it is impossible to demonstrate
unless the parallels exist. So it turns out that those who reason thus
merely say a particular thing is, if it is: in this way everything will
be self-evident. But that is impossible.

If then it is uncertain whether A belongs to C, and also whether
A belongs to B, and if one should assume that A does belong to B, it is
not yet clear whether he begs the original question, but it is evident
that he is not demonstrating: for what is as uncertain as the question
to be answered cannot be a principle of a demonstration. If however B is
so related to C that they are identical, or if they are plainly convertible,
or the one belongs to the other, the original question is begged. For one
might equally well prove that A belongs to B through those terms if they
are convertible. But if they are not convertible, it is the fact that they
are not that prevents such a demonstration, not the method of demonstrating.
But if one were to make the conversion, then he would be doing what we
have described and effecting a reciprocal proof with three
propositions.

Similarly if he should assume that B belongs to C, this being as
uncertain as the question whether A belongs to C, the question is not yet
begged, but no demonstration is made. If however A and B are identical
either because they are convertible or because A follows B, then the question
is begged for the same reason as before. For we have explained the meaning
of begging the question, viz. proving that which is not self-evident by
means of itself.

If then begging the question is proving what is not self-evident
by means of itself, in other words failing to prove when the failure is
due to the thesis to be proved and the premiss through which it is proved
being equally uncertain, either because predicates which are identical
belong to the same subject, or because the same predicate belongs to subjects
which are identical, the question may be begged in the middle and third
figures in both ways, though, if the syllogism is affirmative, only in
the third and first figures. If the syllogism is negative, the question
is begged when identical predicates are denied of the same subject; and
both premisses do not beg the question indifferently (in a similar way
the question may be begged in the middle figure), because the terms in
negative syllogisms are not convertible. In scientific demonstrations the
question is begged when the terms are really related in the manner described,
in dialectical arguments when they are according to common opinion so
related.

Part 17

The objection that 'this is not the reason why the result is false',
which we frequently make in argument, is made primarily in the case of
a reductio ad impossibile, to rebut the proposition which was being proved
by the reduction. For unless a man has contradicted this proposition he
will not say, 'False cause', but urge that something false has been assumed
in the earlier parts of the argument; nor will he use the formula in the
case of an ostensive proof; for here what one denies is not assumed as
a premiss. Further when anything is refuted ostensively by the terms ABC,
it cannot be objected that the syllogism does not depend on the assumption
laid down. For we use the expression 'false cause', when the syllogism
is concluded in spite of the refutation of this position; but that is not
possible in ostensive proofs: since if an assumption is refuted, a syllogism
can no longer be drawn in reference to it. It is clear then that the expression
'false cause' can only be used in the case of a reductio ad impossibile,
and when the original hypothesis is so related to the impossible conclusion,
that the conclusion results indifferently whether the hypothesis is made
or not. The most obvious case of the irrelevance of an assumption to a
conclusion which is false is when a syllogism drawn from middle terms to
an impossible conclusion is independent of the hypothesis, as we have explained
in the Topics. For to put that which is not the cause as the cause, is
just this: e.g. if a man, wishing to prove that the diagonal of the square
is incommensurate with the side, should try to prove Zeno's theorem that
motion is impossible, and so establish a reductio ad impossibile: for Zeno's
false theorem has no connexion at all with the original assumption. Another
case is where the impossible conclusion is connected with the hypothesis,
but does not result from it. This may happen whether one traces the connexion
upwards or downwards, e.g. if it is laid down that A belongs to B, B to
C, and C to D, and it should be false that B belongs to D: for if we eliminated
A and assumed all the same that B belongs to C and C to D, the false conclusion
would not depend on the original hypothesis. Or again trace the connexion
upwards; e.g. suppose that A belongs to B, E to A and F to E, it being
false that F belongs to A. In this way too the impossible conclusion would
result, though the original hypothesis were eliminated. But the impossible
conclusion ought to be connected with the original terms: in this way it
will depend on the hypothesis, e.g. when one traces the connexion downwards,
the impossible conclusion must be connected with that term which is predicate
in the hypothesis: for if it is impossible that A should belong to D, the
false conclusion will no longer result after A has been eliminated. If
one traces the connexion upwards, the impossible conclusion must be connected
with that term which is subject in the hypothesis: for if it is impossible
that F should belong to B, the impossible conclusion will disappear if
B is eliminated. Similarly when the syllogisms are negative.

It is clear then that when the impossibility is not related to
the original terms, the false conclusion does not result on account of
the assumption. Or perhaps even so it may sometimes be independent. For
if it were laid down that A belongs not to B but to K, and that K belongs
to C and C to D, the impossible conclusion would still stand. Similarly
if one takes the terms in an ascending series. Consequently since the impossibility
results whether the first assumption is suppressed or not, it would appear
to be independent of that assumption. Or perhaps we ought not to understand
the statement that the false conclusion results independently of the assumption,
in the sense that if something else were supposed the impossibility would
result; but rather we mean that when the first assumption is eliminated,
the same impossibility results through the remaining premisses; since it
is not perhaps absurd that the same false result should follow from several
hypotheses, e.g. that parallels meet, both on the assumption that the interior
angle is greater than the exterior and on the assumption that a triangle
contains more than two right angles.

Part 18

A false argument depends on the first false statement in it. Every
syllogism is made out of two or more premisses. If then the false conclusion
is drawn from two premisses, one or both of them must be false: for (as
we proved) a false syllogism cannot be drawn from two premisses. But if
the premisses are more than two, e.g. if C is established through A and
B, and these through D, E, F, and G, one of these higher propositions must
be false, and on this the argument depends: for A and B are inferred by
means of D, E, F, and G. Therefore the conclusion and the error results
from one of them.

Part 19

In order to avoid having a syllogism drawn against us we must take
care, whenever an opponent asks us to admit the reason without the conclusions,
not to grant him the same term twice over in his premisses, since we know
that a syllogism cannot be drawn without a middle term, and that term which
is stated more than once is the middle. How we ought to watch the middle
in reference to each conclusion, is evident from our knowing what kind
of thesis is proved in each figure. This will not escape us since we know
how we are maintaining the argument.

That which we urge men to beware of in their admissions, they ought
in attack to try to conceal. This will be possible first, if, instead of
drawing the conclusions of preliminary syllogisms, they take the necessary
premisses and leave the conclusions in the dark; secondly if instea